[Tex/LaTex] How to limit the range of a function in TikZ

tikz-pgf

I have code like this:

\documentclass[tikz]{standalone}

\begin{document}
\begin{tikzpicture}
    \draw [->,thick] (-5,0) -- (5,0) node[right] {$x$};
    \draw [->,thick] (0,-5) -- (0,5) node[above] {$y$};
    \draw[ultra thick, domain=-5:5] plot (\x, {pow(\x,2)-5});
\end{tikzpicture}
\end{document}

When I draw this, the parabola is extremely exceeding the area of what I want. Is there a way to limit the range of the function f(x)=x^2-5 to suit the coordinate area, i.e. only show the points which y-coordinates are between -5 and 5?

Best Answer

You could add a \clip before you do the plotting:

\documentclass[tikz, border=5mm]{standalone}

\begin{document}
  \begin{tikzpicture}
    \draw [->,thick] (-5,0) -- (5,0) node[right] {$x$};
    \draw [->,thick] (0,-5) -- (0,5) node[above] {$y$};
    \clip (-5,-5) rectangle (5,5);
    \draw[ultra thick, domain=-5:5] plot (\x, {pow(\x,2)-5});
  \end{tikzpicture}
\end{document}

Introduction

This is an old question, but all previous answers have limitations: the main one is that all use plot. And plot command produce multiple cubic curves. But to draw a parabola a single quadratic (cubic) curve is enough.

Some explanations

Any parabola can be drawn by a quadratic Bézier curve, and so by a cubic Bézier curve.
(A cubic curve with control points A,B,C,D draws a quadratic one iff AD=3BC.)

The "standard" parabola t(1-t) over [0,1] can be drawn by \draw (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0);.

Every parabola between two points can be obtained by an affine transform from this "standard one". Using this we can define a style parabola through that use a single Bézier curve to draw the desired parabola. This style can be used with to or edge in the following way (A) to[parabola through={(B)}] (C).

The code

The definition of the parabola through is:

\makeatletter
\def\pt@get#1#2{
  \tikz@scan@one@point\pgfutil@firstofone#2\relax%
  \csname pgf@x#1\endcsname=\pgf@x%
  \csname pgf@y#1\endcsname=\pgf@y%
}
\tikzset{
  parabola through/.style={
    to path={{[x={(\pgf@xc,\pgf@yc)}, y=\parabola@y, shift=(\tikztostart)]
      -- (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0) \tikztonodes}--(\tikztotarget)}
  },
  parabola through/.prefix code={
    \pt@get{a}{(\tikztostart)}\pt@get{b}{#1}\pt@get{c}{(\tikztotarget)}%
    \advance\pgf@xb by-\pgf@xa\advance\pgf@yb by-\pgf@ya%
    \advance\pgf@xc by-\pgf@xa\advance\pgf@yc by-\pgf@ya%
    \pgfmathsetmacro\parabola@y{(\pgf@yc-\pgf@xc/\pgf@xb*\pgf@yb)%
      /(\pgf@xb-\pgf@xc)*\pgf@xc}%
  }
}
\makeatother

Note: We can avoid \makeatletter/\makeatother and all @s by using let from the calc library.

We can use (A) to[parabola through={(B)}] (C):

in every case where the parabola exists, so when the three x-coordinates are different,
the point B can be outside the drawn are,
this can be part of a general path with nodes positioned on it.

Example 1:

\tikz\draw[help lines] (0,0) grid (4,3)
  (0,0) edge[parabola through={(3,2)},
    red,thick,fill=blue,fill opacity=.21] (4,1);

Example 2 (Full MWE):

\documentclass[tikz,border=7pt]{standalone}
\makeatletter
\def\pt@get#1#2{
  \tikz@scan@one@point\pgfutil@firstofone#2\relax%
  \csname pgf@x#1\endcsname=\pgf@x%
  \csname pgf@y#1\endcsname=\pgf@y%
}
\tikzset{
  parabola through/.style={
    to path={{[x={(\pgf@xc,\pgf@yc)}, y=\parabola@y, shift=(\tikztostart)]
      -- (0,0) .. controls (1/3,1/3) and (2/3,1/3) .. (1,0) \tikztonodes}--(\tikztotarget)}
  },
  parabola through/.prefix code={
    \pt@get{a}{(\tikztostart)}\pt@get{b}{#1}\pt@get{c}{(\tikztotarget)}%
    \advance\pgf@xb by-\pgf@xa\advance\pgf@yb by-\pgf@ya%
    \advance\pgf@xc by-\pgf@xa\advance\pgf@yc by-\pgf@ya%
    \pgfmathsetmacro\parabola@y{(\pgf@yc-\pgf@xc/\pgf@xb*\pgf@yb)%
      /(\pgf@xb-\pgf@xc)*\pgf@xc}%
  }
}
\makeatother
\begin{document}
  \begin{tikzpicture}
    \draw[help lines] (-1,-1) grid (3,3);
    % variations of the point "through"
    \foreach \y in {-1,-.9,...,1}
      \draw[green] (-1,1) node[black]{.}
        to[parabola through={(0,\y)}] node[black]{.}
          node[black,at end]{.} (1,.5);
    % variations of a boundary point
    \foreach \y in {1.5,1.7,...,3}
      \draw[purple] (-1,2) node[black]{.}
        to[parabola through={(0,2)}] node[black]{.}
          node[black,at end]{.} (1,\y);
    % variations of a point "trough" outside the drawn part
    \foreach \y in {-1,-0.5,...,3}{
      \draw[red,thick] (.5,1) node[black]{.}
        to[parabola through={(3,\y)}] node[black]{.}
          node[black,at end]{.} (2,1);
      \draw[dashed,blue] (.5,1) node[black]{.}
        to[parabola through={(2,1)}] node[black]{.}
          node[black,at end]{.} (3,\y);
    }
  \end{tikzpicture}
\end{document}

Compared to the built in parabola operation

TikZ provide a parabola path operation. But it is not very well designed :

the (0,0) parabola (1,1) is supposed to draw the parabola t^2 between 0 and 1. It draws a cubic curve that is close to this parabola but it is not exactly the same, actually it draws (0,0) .. controls (.5,0) and (0.8875,0.775) .. (1,1), but the exact curve is (0,0) .. controls (1/3,0) and (2/3,1/3) .. (1,1) (not clear why this curve is not used),
when used with bend option, it use two cubic curves to approximate the parabola, but only one is enough to draw the exact one,
when used with bend=<point> option, if you do not choose well the point the curve is not a parabola.

There is a situation where the original parabola is simpler to use (even if not exactly a parabola is drawn), when the bend (the extremal point) is at the start or at the end : (0,0) parabola (2,4) is simpler than (0,0) to[parabola through={(1,1)}] (2,4).

[Tex/LaTex] Numerical conditional within tikz keys

I am definitely unfamiliar with both beamer and tikz (do not quite get what the \only are supposed to do) but perhaps this could go in the direction you want:

\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{chains}
\newcounter{count}

% helper macro:
\long\def\GobToSemiColon #1;{}

\newcommand\myPicture{
\begin{tikzpicture}
  \begin{scope}[start chain = going below]
  \ifnum\value{count}<1 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>3 \expandafter\GobToSemiColon\fi
  \node[draw, rectangle, on chain] {display only when counter is between
    1 and 3};

  \ifnum\value{count}>-1 \expandafter\GobToSemiColon\fi
  \node[draw, rectangle, on chain] {display only when counter is
    negative}; 

  \ifnum\value{count}<100 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>200 \expandafter\GobToSemiColon\fi  
  \node[draw, rectangle, on chain] {display only if counter is between
    100 and 200};

  \ifnum\value{count}<3 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>20 \expandafter\GobToSemiColon\fi
  \node[draw, circle, on chain] {only when counter is in the range 3 to 20};
  \end{scope}
\end{tikzpicture}
}

\begin{document}
\begin{frame}
\only{\setcounter{count}{-3}\myPicture}
\only{\setcounter{count}{105}\myPicture}
\only{\setcounter{count}{39}\myPicture}
\only{\setcounter{count}{2}\myPicture}
\only{\setcounter{count}{5}\myPicture}
\end{frame}
\end{document}

beamer

Best Answer

Related Solutions

[Tex/LaTex] Macro to draw a parabola with pgf/TikZ

Introduction

Some explanations

The code

Compared to the built in parabola operation

[Tex/LaTex] Numerical conditional within tikz keys

Related Question